MOTION MOUNTAIN

80 2 the description of electromagnetic field evolution
Field lines
imagined as
water flow
paddle-wheel
F I G U R E 46 Visualizing the curl of a vector field. Imagine the field to be flowing air and check whether
the small paddle-wheel rotates; if it does, the local curl is non-zero. The direction of the curl is the
direction of the paddle-wheel axis that yields the highest rotation velocity.
Ref. 44
Challenge 78 ny
Ref. 45
Challenge 79 ny
step is the definition of the(magnetic)vector potential, which describes the momentum
per charge that the field provides:
A= p q . (34)
When a charged particle moves through a magnetic potentialA(x), its momentum
changes byqΔA; it changes by the difference between the potential values at the start
and end points, multiplied by its charge. Owing to this definition, the vector potential
has the property that
B=∇×A=curlA (35)
i.e., that the magnetic field is thecurl of the magnetic potential. In most other languages
the curl is called therotation and abbreviated rot. To visualize what the curl or rotation
is, imagine that the field vectors are the velocity vectors of flowing air. Now put a tiny
paddle-wheelatapoint,asshowninFigure46.Ifitturns,thecurlisnon-zero.Therotation
speedof the paddle-wheel is maximal for somedirection of theaxis; this maximal
speeddefines boththemagnitude and thedirection of thecurl at thepoint. (Therighthandrule
is implied.) For example, thecurl for the velocities of a rotating solid body is
everywhere2ω, or twice the angular velocity.
The vector potential for a long straight current-carryingwireisparalleltothewire;it
hasthemagnitude
A(r)=− μ 0I
4π lnr r 0
, (36)
whichdependsontheradialdistancerfromthewireandanintegrationconstantr 0 .This
expressionforthe vector potential, pictured in Figure 45, shows how the moving current
produces a linear momentum in the (electro-)magnetic fieldaroundit. In thecaseof a
solenoid,the vector potential ‘circulates’ around the solenoid.The magnitude obeys
A(r)=− Φ 1
4πr , (37)
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net